The convergence requirement in Assumption 3 is a standard high-level
condition needed for the validity of subsampling procedures
(see [1]}).
Ensuring that Assumption 3 holds in the GAN setting turns out to be
particularly challenging and we resort to empirical process theory
for this. (A thorough account of the needed empirical process theory
can be found in the monograph of [2]}.)
With the set \(\Theta \)  compact as before, let \(l^{\infty }(\Theta )\)  denote
the space of all bounded real-valued functions on \(\Theta \)  equipped
with the supremum norm and \(C(\Theta )\)  stand for the subspace of
those functions that are continuous. Random functions such as \(\hat{f}_{n}(\cdot )\) 
are viewed as maps from appropriate underlying
probability spaces to \(l^{\infty }(\Theta )\) , and \(\rightsquigarrow \) 
denotes weak convergence as defined in [2]}.
